Q. 57

Question

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.

$\underset{x \to \infty}{l i m} \frac{3 x}{x + 1} = 3, ε = 0.5, find smallest N > 0$

Step-by-Step Solution

Verified

Answer

The required value of $N = 5$

1Step 1. Given Information

The given function is $f (x) = \frac{3 x}{x + 1}$

2Step 2. Explanation

From the given function, we have, $c = \infty, L = 3$

The limit expression can be written as a formal statement as below,

For all epsilon positive, there is some N positive such that if $x \in (N, \infty)$

Then $\frac{3 x}{x + 1} \in (3 - ε, 3 + ε)$

Now the smallest value of N is given by,

$\frac{3 x}{x + 1} = 3 - 0.5 \frac{3 x}{x + 1} = 2.5 3 x = 2.5 x + 2.5 0.5 x = 2.5 x = 5$

Q. 56

Q. 58

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Q. 56

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of

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Q. 58

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given val

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Q. 59

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given val

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